SLAC-PUB-17494, FERMILAB-PUB-19-619-SCD

Lepton-Nucleus Cross Section Measurements for DUNE with the LDMX Detector

Artur M. Ankowski,1 Alexander Friedland,1 Shirley Weishi Li,1

Omar Moreno,1 Philip Schuster,1 Natalia Toro,1 and Nhan Tran2

1SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA2Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

(Dated: April 5, 2020)

We point out that the LDMX (Light Dark Matter eXperiment) detector design, conceived to searchfor sub-GeV dark matter, will also have very advantageous characteristics to pursue electron-nucleusscattering measurements of direct relevance to the neutrino program at DUNE and elsewhere. Thesecharacteristics include a 4-GeV electron beam, a precision tracker, electromagnetic and hadroniccalorimeters with near 2π azimuthal acceptance from the forward beam axis out to ∼40◦ angle,and low reconstruction energy threshold. LDMX thus could provide (semi)exclusive cross sectionmeasurements, with detailed information about final-state electrons, pions, protons, and neutrons.We compare the predictions of two widely used neutrino generators (genie, gibuu) in the LDMXregion of acceptance to illustrate the large modeling discrepancies in electron-nucleus interactionsat DUNE-like kinematics. We argue that discriminating between these predictions is well withinthe capabilities of the LDMX detector.

I. INTRODUCTION

The discovery of neutrino masses and flavor mixingrepresents a breakthrough in the search for physics be-yond the Standard Model. As the field of neutrinophysics enters the precision era, accelerator-based neu-trino oscillation experiments are taking center stage.This includes NOvA, T2K, and MicroBooNE, which arecurrently taking data, SBND and ICARUS detectors,which will soon be deployed at Fermilab, and the DeepUnderground Neutrino Experiment (DUNE), for whichthe technical design is being finalized.

The primary goal of the accelerator-based neutrinoprogram is the measurement of oscillation features ina reconstructed neutrino-energy spectrum. Performingthis reconstruction accurately and consistently for bothneutrinos and antineutrinos requires a detailed under-standing of how (anti)neutrinos interact with nuclei—a subtlety that has already impacted past oscillationfits [1–3], despite the availability of near detectors, whichcan help tune cross section models and constrain othersystematic effects. The situation will be even more chal-lenging at DUNE [4], where the science goal is to measurethe subtle effects of δCP and mass hierarchy, requiring amuch higher level of precision.

The origin of these difficulties stems from the com-plexity of neutrino-nucleus interactions in the relevantenergy range, which for DUNE is approximately between500 MeV and 4 GeV. At these energies, different mech-anisms of interaction yield comparable contributions tothe cross section (see Appendix C for details). One hasto model both quasielastic (QE) scattering, in which astruck nucleon remains unbroken, νµ + n → µ− + p,and various processes in which one or more pions areproduced. The latter can occur through the excita-tion of baryonic resonances, as well as through non-resonant channels. At sufficiently high values of four-momentum transfer, Q2 = −(pν−pµ)2, and energy trans-

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FIG. 1. Simulated event distribution for charged-currentmuon neutrino scattering on argon in the DUNE near de-tector, shown as a heat map, compared with the kinematicsaccessible in inclusive and (semi)exclusive electron scatteringmeasurements at LDMX. Blue lines correspond to constantelectron-scattering angles of 40◦, 30◦, and 20◦. Green linesrepresent contours of constant transverse electron momentapT of 800, 400, and 200 MeV. As currently envisioned, LDMXcan probe the region with θe < 40◦ and pT > 10 MeV (belowthe scale of the plot).

fer, ω = Eν − Eµ, the deep inelastic scattering (DIS)description of the interaction becomes appropriate, inwhich the lepton scatters on individual quarks inside thenucleon, followed by a process of “hadronization.”

As DUNE uses argon as a target, all this happensinside a large nucleus, adding further complexity. Thepresence of the surrounding nucleons means hadrons cre-ated at the primary interaction vertex may undergo large

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final-state interactions (FSI) on their way out. The re-sulting intranuclear cascade can lead to energy loss orabsorption of primary hadrons, production of additionalhadrons, and nucleon knockout. Initial states of the in-teracting nucleons are also affected, by nuclear bindingand motions inside a nucleus. Last but not least, multin-ucleon effects, such as meson-exchange currents (MEC),which arise from scattering on interacting nucleon pairs,likewise have to be considered.

To model this rich physics, experiments rely on eventgenerator codes, among them genie [5, 6] and gibuu [7–11], which are used as benchmarks in this paper. Aswe will see explicitly below, these codes are often not inagreement with each other. More importantly, they areoften also not in agreement with recent high-statisticsdata from the MINERvA experiment, collected in thekinematic regime relevant to DUNE. For example, the de-fault models in genie seem to significantly overestimateneutron production [12], mispredict the ratio of charge-current interactions across different nuclear targets [13],and mismodel single-pion production [14]. Thus, there isdirect experimental evidence that existing models needto be improved.

Importantly, simple phenomenological tuning of pa-rameters within the existing models may not be suffi-cient. For example, Ref. [14] reports that no tune coulddescribe all different exclusive final states in their anal-ysis. Crucially, the paper also notes that the physicalorigin of the discrepancies is difficult to pinpoint, basedon only the available data.

This brings us to an important question: what newdata are needed to improve the physics in these genera-tors? A priori, one might think that all that is needed ismore neutrino-nucleus scattering data, with higher statis-tics and precision, as will be collected with the futurenear detectors. In reality, while better neutrino datawould certainly be desirable, it is unlikely to be suffi-cient. To date, neutrino experiments only have accessto broadband beams, extract flux-integrated cross sec-tions [15–23], and neutrino-energy reconstruction itselfsuffers from sizable uncertainties. In turn, the processof energy reconstruction relies on neutrino generators.The reason is that even today’s state-of-the-art neutrinodetectors are imperfect calorimeters at several GeV ener-gies, with event generators being used to fill in the miss-ing information. Hence, complementary probes that arefree from these limitations are highly desirable for accu-rately validating the physical models in event generators.

Precise electron-nucleus scattering data provide justsuch a complementary probe. While electron and neu-trino interactions are different at the primary vertex,many relevant physical processes in the nucleus are thesame in the two cases, as discussed below in Sec. II. Whatelectron scattering offers is precisely controlled kinemat-ics (initial and final energies and scattering angles), largestatistics, in situ calibration of the detector response us-ing exclusive reactions, and a prospect of easily swappingdifferent nuclear targets. This allows one to easily zero in

on specific scattering processes and to diagnose problemsthat are currently obscured by the quality of the neutrinoscattering data.

In this paper, we point out that the proposed LDMX(Light Dark Matter eXperiment) setup at SLAC [24],designed to search for sub-GeV dark matter, will havevery advantageous characteristics to also pursue electron-scattering measurements relevant to the neutrino pro-gram. These include a 4-GeV electron beam and a de-tector with high acceptance of hadronic products in the∼40◦ forward cone and low-energy threshold. Figure 1shows the distribution, in the (ω,Q2) plane, of charged-current (CC) events for muon neutrino scattering on ar-gon nuclei in the near detector of DUNE, simulated withthe gibuu generator code. As can be immediately seen,the LDMX coverage in the relevant kinematic windowis excellent. Below, we quantify how future LDMX datacan be used to test and improve physics models in lepton-nucleus event generator codes.

II. ELECTRON SCATTERINGMEASUREMENTS AND NEUTRINO CROSS

SECTIONS

Let us now define the connection between electron- andneutrino-nucleus scattering more precisely. Superficially,the mere existence of such a connection is not obvious,since the weak and electromagnetic forces have a num-ber of important differences. The differences are imme-diately apparent in the elastic scattering regime: whileCC neutrino interactions occur on initial-state neutronsin the nucleus, electromagnetic scattering also involvesinitial-state protons (neutrons couple through their mag-netic moments). The situation is similar in the DISregime, where the primary vertex is treated at the quarklevel: while CC neutrino (antineutrino) interactions arecontrolled by the distribution of initial-state down (up)quarks, electron scattering involves both up and downquarks. Additional differences come from the chiral na-ture of the weak interactions. While the electron-nucleonvertex is sensitive only to the electric charge distribu-tion inside a nucleon and its magnetic moment, neutrinoscattering also depends on the distribution of the axialcharge. The effect of this axial coupling is not small; infact, at 1-GeV neutrino energy, the axial part of the weakinteraction provides a dominant contribution to the elas-tic neutrino-nucleus cross section. In short, one shouldnot expect to blindly convert electron-scattering datainto predictions for neutrinos.

Yet, a tight connection between electron and neutrinoscattering does exist. This is most immediately seen byconsidering the nuclear physics of the problem. Neutrinoscattering depends on the wave functions of the initialnucleons (in momentum space) and on the nuclear den-sity profile, and these are most accurately probed withelectron scattering. The differences between proton andneutron couplings mentioned above might give one pause.

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However, by systematically analyzing electron data onmirror nuclei, in which neutrons in one nucleus have thesame shell structure as protons in another, one can learnabout both proton and neutron wave functions [25–27].

The same argument can be made about modelingfinal-state interactions, which dictate the subsequentevolution of the interaction products inside the nu-cleus [7, 28, 29]. FSI can significantly modify the proper-ties of the hadronic system, through energy loss of prop-agating particles, absorption and creation of mesons, aswell as nucleon knockout. It is essential to model theintranuclear transport of various hadrons using a uni-fied framework, regardless of whether they are producedin electron or neutrino scattering. The accuracy of thetreatment can then be validated by targeted studies ofexclusive hadronic final states in electron scattering.

We see here that it is important for the differences be-tween electron and neutrino interactions to be limited tothe elementary scattering vertex. This is justified at typ-ical momentum-transfer values relevant to DUNE, wherescattering involves predominantly a single nucleon. Itcan be shown that, under these conditions, nuclear ef-fects become largely independent of the interaction dy-namics [30].

The connections between electron and neutrino scat-tering, in fact, extend beyond nuclear physics models,to include many hadronic physics effects. For example,to model neutrino-quark interactions in the DIS regime,one needs accurate parton distribution functions. Thesecan be extracted from precision electron-scattering data.The physics of the subsequent hadronization can also betreated in a common framework. Finally, it is desirable touse a unified treatment of other physics, such as hadronicresonances, two-nucleon currents, or quark-hadron dual-ity. Of course, in doing so, one needs to include thecorrect treatment of the nucleon axial properties. Eventhere, however, comparisons to electron scattering areproving to be highly advantageous. For example, re-cent lattice QCD studies found it useful to simultane-ously model the nucleon axial and vector form factors(see, e.g., Refs. [31–35]).

The importance of using the same nuclear model forneutrino and electron scattering was realized a long timeago, as illustrated, for instance, by the discussion in theseminal paper by Smith and Moniz [36]. In fact, it was ar-gued in that paper that combining electron and neutrinoscattering gives one the best tool for probing the physicsof the nucleus. The same argument has also been mademore recently from the experimental point of view [37].It has since been incorporated into the mission state-ment of the GENIE generator. Insofar as this crucialprinciple is adhered to in the generator development andapplications, electron-scattering data should provide anexcellent validation platform.

Let us next outline the requirements from the point ofview of neutrino experiments. As stated in the Introduc-tion, the key to many modern neutrino experiments is ac-curate neutrino-energy reconstruction. Experiments such

as NOvA and DUNE approach this problem by using thecalorimetric technique, which involves adding up visibleenergies of all final-state particles and inferring invisiblecomponents, such as neutrons and low-energy chargedhadrons, using event-generator predictions. Event gen-erators are also used to model the composition of thefinal-state hadronic system, whenever that informationis unavailable from the particle-identification algorithms.Knowledge of the final-state composition is needed toconvert measured ionization charge, or scintillation light,to true energy loss. This is not a small effect, and exist-ing differences among generator models consistent withavailable validation data can yield energy reconstructionvariations as large as 20%, which has been discussed sys-tematically in Ref. [38], together with other factors im-pacting the energy resolution.

Thus, to adequately constrain the underlying gener-ator models, one needs to measure not only inclusiveelectron-scattering rates, but also collect detailed infor-mation about the exclusive hadronic final states. Thisincludes charged pions, neutral pions, and protons, aswell as any available information on final-state neu-trons. Practically, one needs to simultaneously mea-sure the kinematics of an energetic, often-forward elec-tron, as well as detect charged hadrons to below 100–200 MeV momenta (see, e.g., Ref. [39]) with wide andwell-characterized angular acceptance.

Discrepancies between scattering data and generatorpredictions can indicate problems either with the nuclearmodel or with hadronic physics [40, 41]. Having informa-tion on exclusive hadronic final states can help diagnosethe origin of the problem. To conclusively disentangle nu-clear and hadronic effects may require comparative anal-yses of electron-scattering data on various nuclear tar-gets, including the lightest elements—helium, deuterium,and hydrogen. That such targets can be quite small inthe case of electron scattering represents another tangi-ble advantage over neutrino scattering, where concernsabout fire safety make future hydrogen bubble-chamberexperiments prohibitively costly.

To this end, a systematic analysis of the data collectedon various nuclear targets by different experiments us-ing the CLAS detector in Hall B at Jefferson Laboratory,while not completely addressing the requirements out-lined above, would be an important advance. So far, thepublished studies focused on specific hadronic processeswith hydrogen targets [42–49]. These should already beuseful for testing generator models for certain hadronicprocesses, such as ρ meson production through higherresonances. The CLAS12 proposal “Electrons for Neu-trinos” would make further inroads by collecting moredata [50, 51]. At present, published datasets involv-ing argon and its mirror nucleus titanium come from aseparate experiment in Hall A [27, 52, 53]. While un-doubtedly valuable [54–58]—for example, enabling com-parisons with the well-studied carbon data [40]—they arelimited to the inclusive spectrum of scattered electronsmeasured at a single value of the beam energy (2.22 GeV)

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FIG. 2. Schematic of the LDMX experiment for dark-mattersearch (not to scale). The electron beam is incident from theleft and interacts in the target (which can be varied). Directtracking and calorimetry along the beam axis provides excel-lent (nearly 2π azimuthal) forward acceptance to a range offinal-state particles, including the recoiling electron, protons,pions, and neutrons.

and a fixed scattering angle (15.54◦).At the moment, and over the next several years, elec-

tronuclear scattering data with excellent hadronic final-state reconstruction is sorely needed. The ideal would bereconstruction with no detection threshold, full 4π cov-erage, and with excellent neutron identification. WhileCLAS12 can make some inroads in this direction, its ac-ceptance will be limited (especially in the forward direc-tion) and neutron-energy reconstruction will be modest.The proposed LDMX detector concept offers a numberof complementary and unique advantages that can beleveraged to provide a range of valuable electron-nucleusscattering data for the purpose of constraining neutrino-scattering models.

III. THE LDMX DETECTOR CONCEPT ANDELECTRON-NUCLEUS SCATTERING DATA

LDMX is a fixed-target experiment designed to searchfor sub-GeV dark matter, employing a high-repetitionrate, low-current electron beam [24] with precision track-ing (in a magnetic field) and calorimetry along the beamaxis to provide high-fidelity detection of both chargedand neutral particles. Figure 2 provides a high-level il-lustration of the detector layout, which is largely opti-mized to search for dark-matter production. In candi-date events for dark-matter production, most of the ini-tial electron’s energy is expected to be carried away byundetected particle(s). Therefore, identification of theseprocesses requires an excellent hermeticity of the detec-tor, allowing, e.g., energetic neutron-knockout events to

be detected with sufficiently small uncertainty. In fact,the primary purpose of the downstream calorimetry inLDMX is to provide a fast, radiation-hard, and highlygranular veto against photonuclear and electronuclear re-actions in the target area that might generate difficult-to-detect final states, and hence a potential backgroundto dark-matter reactions. In the nominal design, the vastmajority of triggered data would be composed of thesephoto/electronuclear reactions, and rejected offline. Thekey result of this paper is that this vetoed data will it-self be of great value in service of neutrino-interactionmodeling, as was described above.

To see why this is the case, we start with a more de-tailed description of the detector layout. The trackingsystem upstream of the target and the target itself arehoused inside of a 1.5-T dipole magnet while the down-stream (recoil) tracker is in the fringe magnetic field. Thetarget is currently envisioned to be titanium, and we as-sume it to be 0.1 X0 (0.356 cm) thick, X0 being the ra-diation length. However, different target materials (suchas argon) and thicknesses are possible, as discussed fur-ther in Sec. VII. The two tracking systems provide robustmeasurements of incoming and outgoing electron momen-tum.

The Electromagnetic Calorimeter (ECal) is surround-ed by the Hadronic Calorimeter (HCal) to provide largeangular coverage downstream of the target area, in or-der to efficiently detect interaction products. The ECalis a silicon-tungsten high-granularity sampling calorime-ter based on a similar detector developed for the high-luminosity Large Hadron Collider upgrade of the endcapcalorimeter of the Compact Muon Solenoid (CMS) detec-tor. The ECal is radiation tolerant with fast readout, andthe high granularity provides good energy resolution andshower discrimination for electromagnetic and hadronicinteractions. The HCal is a scintillator-steel samplingcalorimeter that has wide angular coverage and is suf-ficiently deep to provide required high efficiency for de-tecting minimum ionizing particles and neutral hadrons.

While the final detector design is still under develop-ment, we describe a coarse set of detector capabilities(motivated by the baseline design), which are particu-larly relevant for electron-scattering measurements [24]:

• Electrons: We estimate the electron energy reso-lution to be 5%–10% and the pT resolution to be< 10 MeV [24], where pT is the transverse momen-tum of the outgoing electron. The tracker accep-tance is approximately 40◦ in the polar angle wherethe z-axis is defined along the beamline. Electronscan be measured down to a kinetic energy of ap-proximately 60 MeV.

• Charged pions and protons: The energy and pT res-olutions, tracking acceptance, and kinetic thresh-olds are similar for charged pions, protons, andelectrons. The estimate of tracking angular andmomentum acceptance is shown in Fig. 3. The re-coil tracker and ECal detectors can be used to per-

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form particle identification via mean energy loss(dE/dx) to separate charged pions and protons.Based on previous studies of similar silicon-trackingtechnologies at CMS [59, 60], the recoil tracker byitself has good pion/proton discrimination powerfor kinetic energies < 1.5 GeV.

• Neutrons: The nominal neutron signal is a hadronicshower in the HCal, although the shower canstart in the ECal, which is roughly one hadronic-interaction mean free path in thickness. Thesignature also requires that there be no charged(minimum ionizing particle) track aligned withthe shower in the tracking (ECal) system. Iden-tifying/reconstructing single neutrons will relyon localized and separable hadronic showers.Once identified, neutrons can be efficiently distin-guished from charged hadrons (protons, chargedpions/kaons) at angles < 40◦ by identifying thosecharged tracks in the tracking and ECal detectors.

Based on Geant4 simulations for the baselineHCal sampling fraction, we estimate the HCal tohave an energy resolution for neutrons of 5% ⊕40%/

√E/GeV and a polar angular acceptance of

65◦. However, because we have tracking accep-tance out to ∼40◦, our studies assume that wehave good pion/proton/neutron discrimination outto only ∼40◦ [24]. We have also assumed thatthe angular resolution of the neutrons are conser-vatively 10◦ based on position resolution measure-ments. We leave it to future studies to understandadditional separation power between 40◦ and 65◦.

Of course, detecting a detached cluster in the HCaldoes not guarantee that it was created by a neu-tron that came from the primary electron inter-action vertex. Some neutrons can be created insecondary and tertiary interactions of energeticcharged hadrons. Understanding how well the pri-mary neutron component can be isolated requiresdedicated future simulations.

• Readout rate: The total data acquisition (DAQ)rate of the detector is approximately 5 kHz. Asignificant fraction of the DAQ bandwidth tar-gets high-energy-transfer reactions. Thus, for thisstudy, we focus on electron energy transfer ω >1 GeV. This energy-transfer threshold is still belowthe nominal threshold for the dark-matter search,but could be achieved by prescaling the trigger orby using a combination of ECal and HCal online se-lections. Even smaller values of ω may be possible,but we leave such studies to future work.

For the studies described below, we assume a 4-GeVincoming electron beam and a dataset of 1 × 1014 EoT(electrons on target), corresponding to approximately 6months of data collecting during an envisioned first phaseof low-luminosity running. The beam repetition rate is

FIG. 3. The acceptance of a charged particle (pion) track asa function of its momentum and polar angle. The acceptanceis defined as a charged particle that leaves four hits in therecoil tracking system.

assumed to be 46 MHz and the beam is tuned to have onaverage one electron per bucket.

With the beam and detector configurations describedabove, we will next explore the potential for LDMX toperform measurements of both inclusive (Sec. V) and(semi)exclusive (Sec. VI) electron-nucleus scattering pro-cesses.

IV. MONTE CARLO GENERATORS

We study the modeling of electron-titanium interac-tions using the Monte Carlo generators genie (versions2.12.8 and 3.0.6) [5, 6] and gibuu (versions 2017 and2019) [7, 11]. As both genie and gibuu had major up-dates, we show results obtained using both the versionsbefore and after these changes. In the context of the in-clusive cross sections, we also present the results obtainedusing Geant4 (version 4.10.p3) [61], for reference.genie [5, 6] is the generator most widely used in

neutrino experiments and the default code employed inDUNE studies. In this analysis, we use its default config-urations (“DefaultPlusMECWithNC” for version 2.12 and“EMPlusMEC G18 02a 00 000” for version 3.0). Nucleareffects are described in genie within the global relativis-tic Fermi gas model of Bodek and Ritchie [62]. Thisapproach treats the nucleus as a fragment of noninter-acting nuclear matter of constant density, bound in aconstant potential. The effect of short-range correlationsbetween nucleons is added in an ad hoc manner, by ex-tending the step-function momentum distribution abovethe Fermi momentum, pF ' 240–250 MeV, with a high-momentum tail. The binding energy is taken to be inde-pendent of momentum, and fixed to a value ∼30 MeV.As a consequence, nucleons in the high-momentum tailof the Bodek-Ritchie model are typically unbound.

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Pion production through excitation of nucleon reso-nances is described in genie 2.12 using the frameworkof the Rein-Sehgal model [63]. While the original workincluded 18 resonances and accounted for interference be-tween them, its implementation in genie disregards theeffect of interference, and is limited to 16 resonances,which are described using up-to-date parameters. In ge-nie 3.0, the default model for resonance excitation is theapproach of Berger and Sehgal [64].

All mechanisms of pion production on nucleons thatdo not involve resonance excitation are referred to in ge-nie as DIS processes. They are modeled following theeffective approach of Bodek and Yang [65, 66]. Relyingon leading-order parton-distribution functions [67], thismodel applies higher-order corrections to the effectivemasses of the target and the final state, in order to ex-tend the applicability of the parton model to the low-Q2

region. While DIS is the only mechanism of interaction ingenie for the invariant hadronic masses W ≥ 1.7 GeV, itis also employed to produce nonresonant background ofevents involving one or two pions in the resonance region,corresponding to W < 1.7 GeV.

gibuu [7, 11] is a Monte Carlo code based on transporttheory, originally developed to describe heavy ion colli-sions. Its nuclear model accounts for the nuclear den-sity profile determined in electron scattering accordingto Ref. [68], treating the nucleus as a local relativisticFermi gas, bound by a potential exhibiting momentumdependence [55].

The implementation of both resonance-excitation pro-cesses and single-pion nonresonant background in gibuumakes use of the MAID analysis [69]. MAID includes 13resonances with invariant mass W ≤ 2.0 GeV and ac-counts for the interference between them, as well as forthe interference between the resonant and nonresonantcontributions. The two-pion rate is estimated by gen-eralizing the model in Ref. [70] for photoproduction, byusing the assumptions of Ref. [71].

To describe DIS processes, gibuu relies on a modifica-tion of the pythia code [72], extending its applicabilitydown to the invariant hadronic mass 2.0 GeV. In thismanner, leading order processes are implemented in theprimary interaction vertex.

Performing simulations using the Geant4 genera-tor [61], we rely on the description of electron-nucleusinteractions within the Bertini cascade model [73] withimprovements discussed in Ref. [24]. This model relies ona parametrization of photoproduction data [74] to obtainthe elementary cross sections for electrons by employingthe equivalent-photon approximation [75, 76]. The nu-clear model of Geant4, based on a local nonrelativisticFermi gas model, approximates the density profile of amedium-sized nucleus as three regions of constant den-sity [73]. Within every region, the binding energy of nu-cleons depends on the local Fermi momentum and on theatomic charge and mass numbers. For pions, a universalconstant potential is used.

In order to eliminate trivial differences between the

three Monte Carlo generators, we apply the kinematicselection Q2 > 0.03 GeV2 [77], needed to define a phasespace where all the generators are physically valid1. Thisselection has no visible effect on the presented cross sec-tions, required to pass our trigger selection, ω > 1 GeV,and the cut on the transverse momentum of the scatteredelectron, pT > 200 MeV (cf. Appendix B).

V. INCLUSIVE MEASUREMENTS

In the baseline detector configuration, we study thepotential for LDMX to make measurements of electron-nucleus processes, the results of which can be used toimprove Monte Carlo generators. In this section, we fo-cus on the simplest inclusive measurements LDMX canperform, namely, on the distribution of the scattered elec-trons on the (θe, ω) plane, θe and ω being the scatteringangle and the energy transferred to the nucleus, respec-tively. Until Sec. VI, we do not consider any informa-tion on the composition or kinematics of the final-statehadrons. Here we argue that LDMX will complement theexisting knowledge of the inclusive cross sections from thevery forward direction to larger scattering angles by pro-viding results for large energy transfers, where they arenot available yet [78]; see Appendix A.

Our analysis is focused on the fiducial region of thescattered electron’s phase space defined by ω > 1GeV and pT > 200 MeV. This selection is synergisticto the LDMX dark-matter search. Before performingthese kinematic selections, we apply parametric angularand momentum/energy smearing of electrons, chargedhadrons, and neutral hadrons, according to the expecteddetector resolutions described above. We also apply an-gular acceptance criteria according to the detector ac-ceptance described in Sec. III. Efficiency effects due toparticle identification algorithms are not applied and re-quire further study. However, we expect them to be veryuniform, well measured, and near unity.

Figure 4 illustrates the distribution of energy trans-ferred by electron to the nucleus for two different se-lections on its scattering angle θe, following the com-mon kinematic selections and energy smearing describedabove. In this energy range, all kinematic features inthe simulation are broader than the energy resolution.The presented results correspond to the expected num-ber of events for 1 × 1014 EoT. In the figure, the bandsrepresent statistical uncertainties of the generated MonteCarlo event samples; experimental statistical uncertain-ties are much smaller than the indicated bands.

The event distributions obtained using the three gen-erators differ markedly both in the overall rate and inshape. Exhibiting stronger angular dependence, gibuu

1 As genie generates events relying on Q2 and the cross sectionfor electrons is divergent when Q2 → 0, it is necessary to imposea cut on the minimal Q2 value.

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FIG. 4. Event distribution as a function of electron energy transfer for the scattering angles of 10◦ ≤ θe ≤ 12.5◦ (left panel)and 20◦ ≤ θe ≤ 22.5◦ (right panel). Scattered electrons are required to have the transverse momentum pT > 200 MeV. In ournominal analyses, the trigger selection ω > 1 GeV is employed.

predicts fewer events at large scattering angles than ex-pected according to genie.

This behavior is shown in Fig. 4. In the left panel,corresponding to scattering angles 10◦ ≤ θe ≤ 12.5◦, theprediction of gibuu is smaller by 30% than that of genie.In the right panel, for scattering angles 20◦ ≤ θe ≤ 22.5◦,this difference increases to 50%. While the genie crosssection is dominated by the DIS channel, this is not thecase for the gibuu results, in which resonance excitationis the main mechanism of interaction for energy transfersbelow 2 GeV, and DIS dominates only at ω > 2 GeV.The largest discrepancies occur at higher-energy trans-fers (ω >∼ 2 GeV, W 2 >∼ 4.4 GeV2), where events are pre-dominately populated by DIS. Notably, there are visibledifferences between the results obtained using differentversions of the generators gibuu and genie. Neverthe-less, they are much less significant than the differencesbetween the predictions of different generators.

In the 20◦ ≤ θe ≤ 22.5◦ slice, both the gibuu and ge-nie cross sections result entirely from DIS interactions,and agree at a factor of 2 level. Geant4, however, de-viates significantly from genie and gibuu, and the de-viation is even larger at higher scattering angles. Thisis expected as Geant4 uses the equivalent-photon ap-proximation to simulate electron-nucleus interactions. Athigher Q2, the exchanged photon becomes highly virtualand this approximation is not valid. Because of this is-sue, we do not show Geant4 predictions in later compar-isons. We note that as Geant4 is not commonly used asan event generator, the difference between Geant4 andother generators is not a fair representation of the currentmodeling uncertainty. However, the difference betweengenie and gibuu is, and it may even be a conservativeestimate on modeling uncertainties. Comparably largedisagreements between genie and gibuu are seen in all

angular bins, as illustrated in Appendix B.Notice that, in Fig. 4, the ranges of electron-scattering

angles are narrow and the final energies are well mea-sured. This, combined with precise knowledge of theinitial electron energy, makes it possible to accuratelycontrol the scattering kinematics, which in turn providesa powerful tool for testing the underlying nuclear andhadronic physics. The large discrepancies between thegenerator predictions for the double-differential cross sec-tion seen in the figure may be less pronounced in moreintegrated quantities. We explicitly confirmed this byintegrating the electron-scattering cross sections for a 4-GeV beam energy over all scattering angles and energytransfers (imposing the same Q2 > 0.03 GeV2 cut asbefore). In this case, we find that the predictions of ge-nie and gibuu are, in fact, in good agreement. Bothgenerators give 1.9×10−28cm2, with the underlying dis-crepancies completely washed out upon integration.

One has to be mindful about this when interpretingresults of neutrino-scattering experiments, where averag-ing can take place over several variables, including theincoming beam energy. As an illustration, consider mea-surements of pion production induced by charged-currentneutrino interactions in the MINERvA experiment, atthe kinematics similar to that of DUNE. The shape of thesingle differential dσ/dQ2 cross sections from Ref. [79]is reproduced reasonably well by both genie [79] andgibuu [9].

This clearly illustrates a general point: for the pur-pose of testing the physics models in the generators, de-tailed measurements of multiply differential cross sec-tions are essential. Fortunately, many such measure-ments are already available from MiniBooNE [15], MIN-ERvA [16–20], MicroBooNE [21], and T2K [22, 23], andmore can be expected in the future (exploring various

8

semi-inclusive modes). These data, when combined withthe electron-scattering measurements described here, willprovide a very powerful foundation for generator devel-opment. Note that the electron and neutrino measure-ments are essentially complementary: the electron mea-surements do not suffer from beam-integration effects,while the neutrino measurements are sensitive to the ax-ial current effects.

LDMX will measure inclusive electron-nucleus scat-tering rates for energy transfers 1 <∼ ω <∼ 4 GeV andscattering angles 5◦ <∼ θe <∼ 40◦. With expected > 105

events per bin, the experimental statistical errors will beat subpercent-level. Instrumental systematic uncertain-ties are difficult to assess precisely prior to data taking,but the scales of many effects can be estimated by com-parison to detailed performance studies of other similarcollider and fixed-target experiments. LDMX’s luminos-ity can be precisely measured by counting incident elec-trons in the tagging tracker and measuring the targetthickness. Electron-reconstruction performance can bequantified precisely using standard candle reactions, suchas Møller scattering. Efficiency uncertainties should besmaller and more uniform than the ∼4% level achievedby the less hermetic CLAS detector (see, e.g., Ref. [49]).Momentum resolution uncertainties should be compara-ble to the ∼3% achieved at HPS [80], which has a simi-lar detector geometry and beam. Such resolution wouldlead to negligible systematic effects on the distributionsin Fig. 4, which vary over much larger energy scales.

Based on these considerations, both statistical and sys-tematic uncertainties are expected to be small, comparedwith the current theoretical uncertainties. These featureswill enable LDMX to discern between gibuu, genie, andGeant4 predictions with high precision, and to performmeasurements of the inclusive cross sections for electronscattering on nuclear targets, such as titanium, over abroad kinematics, previously unexplored. Availabilityof such results is essential for future development andtuning of Monte Carlo generators employed in the long-baseline neutrino-oscillation program.

VI. (SEMI)EXCLUSIVE MEASUREMENTS

As explained in Sec. II, Monte Carlo generators playa fundamental role in neutrino-energy reconstruction, re-lating the visible energy—deposited in the detector bythe observed particles—with the actual neutrino energy.In order to do so, the contribution of undetected energy—carried away by undetected particles, absorbed in nuclearbreakups, etc. [38]—is estimated based on the measuredevent composition and kinematics. The accuracy of theenergy reconstruction relies on the accuracy of the par-ticle multiplicities and spectra predicted by the MonteCarlo simulation. Therefore, availability of precise infor-mation on the hadronic final states is essential to vali-date the models underlying the generators and to esti-mate their contribution to the systematic uncertainty of

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energy reconstruction.

Here, we present spectra obtained for coincidence mea-surements in LDMX of electrons, pions (with particleID), and neutrons. We argue that thanks to the angularcoverage of LDMX, the measurements can be performedwith high efficiency across a broad range of energy andangle.

As an example, in Fig. 5 we show the energy fractionthat goes into different hadronic particles when the elec-tron scattering angle is between 20◦ and 22.5◦ and the en-ergy transfer exceeds 1 GeV (corresponding to the rightpanel of Fig. 4). The shaded areas in Fig. 5 illustratethe energy fractions that are outside LDMX acceptance,predominantly due to the angular coverage. We observethat most of the final-state particles are within LDMXacceptance. The neutron acceptance is slightly lower alsodue to the high threshold, the kinetic energy of 500 MeV.Even then, LDMX can detect ∼50% neutrons.

The hadronic energy fractions predicted by a gener-ator depend on the interaction channel dominating itstotal cross section. While nucleons in the final statecarry more energy in the resonance-excitation channelthan in DIS, for pions this situation is reversed. Yieldinga larger resonance contribution to the total cross sectionthan genie, gibuu predicts ∼40% less energy carriedby electromagnetic showers initiated by neutral pions,and more energy carried by neutrons. The latter issueis of particular importance because neutrons are partic-ularly difficult to measure in neutrino detectors. If leftunresolved, such large discrepancies would result in large

9

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FIG. 6. Charged pion kinetic energy distribution after energy/angular resolution smearing with a scattering angle of 0◦ ≤ θπ ≤20◦ (left panel) and 20◦ ≤ θπ ≤ 40◦ (right panel). There is an additional selection on the recoiling electron of ω > 1 GeV,pT > 200 MeV, and 20◦ ≤ θe ≤ 22.5◦. The pion distributions are presented per electron within the above electron kinematicselection. There are approximately 1 × 108 electrons passing the ω, pT , and θe selections for 1 × 1014 electrons on target.

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FIG. 7. Neutron polar angle distribution after energy/angularresolution smearing. There is an additional selection onthe recoiling electron of ω > 1 GeV, pT > 200 MeV, and20◦ ≤ θe ≤ 22.5◦. The neutron distributions are presented perelectron within the above electron kinematic selection. Thereare approximately 1× 108 electrons passing the ω, pT , and θeselections for 1 × 1014 electrons on target.

uncertainties on the inferred neutrino energy [38]. Bymeasuring these hadronic energy fractions within its ge-ometric acceptance, LDMX will provide a good handleon the relative rate of neutron emission.

More specifically, the capability of LDMX to measurein coincidence the kinematics of the scattered electronand of the hadronic interaction products is illustrated bythe distributions shown in Figs. 6 and 7.

Figure 6 presents the pion kinetic energy distribu-tions expected in LDMX when the corresponding elec-tron kinematics is selected in a similar manner as inthe previous section: ω > 1 GeV, pT > 200 MeV, and20◦ ≤ θe ≤ 22.5◦. We expect approximately 1×108 elec-trons with that particular kinematic selection for 1×1014

electrons incident on the target. After accounting for theacceptance and energy resolution of the tracker, LDMXcan measure the charged-pion kinetic energy down to∼60 MeV. We present the distribution up to 1 GeV,where LDMX is expected to have good pion/proton dis-crimination.

The distributions in Fig. 6 are normalized per elec-tron meeting the selection criteria, in order to removethe generator differences for inclusive electron scatteringdiscussed in Sec. V. We see that genie predicts more pi-ons, about a factor of 2 more in the forward region, whilegibuu yields a slightly harder pion spectrum.

Similarly to the electron case, the pion energy resolu-tion is sufficiently small that its effect is invisible in thefigure, and features in pion spectra predicted by gener-ators, e.g., the peak toward lowest pion energies due tofinal-state interactions, are preserved. We also observe asensitivity to the difference between the pion spectra for0◦ ≤ θπ ≤ 20◦ and for 20◦ ≤ θπ ≤ 40◦, illustrating theadvantage of having fine-grained tracking detector for allcharged particles.

In Fig. 7, the angular distributions of all neutronsin an event within the acceptance of the tracker andcalorimeter and with (smeared) kinetic energies greaterthan 500 MeV are shown. Again, this is with the sameselection on the electron as in the pion result. The dis-tributions show large overall rate differences between thegenerators, but even within the shape of the distribu-tions, there are differences at the 30%–40% level.

10

From the representative distributions we have shownfor the electron and hadron kinematics, it is clear thatthere are large deviations in the predictions of electron-nucleus interactions from various state-of-the-art gener-ators. LDMX will provide good measurements of thesemultiparticle final states. Figures 6 and 7 show the pionkinetic energies and neutron angular distributions per in-coming electron within a narrow angular slice, but as isnoted above, we expect approximately 1 × 108 electronswith that kinematic selection. Therefore, the per-bin sta-tistical uncertainties on these measurements will be atthe percent level or smaller.

The systematic uncertainties discussed in the contextof inclusive measurements translate directly to the caseof (semi)exclusive measurements. The main new system-atic in this case is the efficiency and cross-contaminationof hadron particle identification using dE/dx. For 1 GeVand below, the rate of cross-contamination for charged pi-ons and protons is likely to be similar to the several per-cent level observed at CMS [59, 60]; this sets the scale fora conservative estimate of the systematic uncertainties aswell. Contamination from kaons, due to their much lowerabsolute rate, is expected to be even less than from pro-tons and pions. For neutron identification, the detectortechnology chosen, scintillator-based sampling calorime-try, is quite mature. While the readout technology andgeometry is different, the CMS experiment measures neu-tral hadrons down to the GeV scale and the uncertaintieson energy measurements are at the ∼10%–20% level [81].

To summarize, similarly to the inclusive case, the ex-pected statistical and systematic errors will be suffi-ciently small to enable precise measurements of (semi)-exclusive electron-nucleus cross sections, by detectingfinal-state hadrons in coincidence with scattered electron.This data will be vital to understanding neutrino-nucleusinteractions and event reconstruction at DUNE. Further-more, it is important to note that there is very little ex-isting data for exclusive measurements of neutron knock-out induced by electron-nucleus scattering, and thus, anysuch measurements will be important to constrain MonteCarlo models.

VII. FUTURE POTENTIAL

In the baseline dark-matter configuration and nomi-nal running, LDMX can be expected to perform valuablemeasurements of both inclusive and (semi)exclusive elec-tron scattering on nuclear targets of interest for DUNE.Here we enumerate potential ways, some more challeng-ing to realize than others, to extend the physics programbeyond the nominal one:

• The nominal physics selections can be extendedto smaller energy transfers to fully cover the re-gions in which resonance-production and meson-exchange currents provide important contributionsto the cross section. However, there are challengeswith triggering on this topology (prescaling is a

possibility) and eventually also issues of detectorresolution. More study is left for future work tounderstand the impact of such measurements.

• In this analysis we assume a 4-GeV electron beam,but there is potential for extending measurementsto higher energies. In particular, an 8-GeV electronbeam from LCLS-II will move the LDMX accep-tance contours to the right in Fig. 1. This wouldallow to cover more of the DIS phase space with rel-atively little change in the detector configuration.

• Varying the target material would provide moredata for nuclear modeling, allowing for deeper un-derstanding of the cross-sections’ dependence onthe atomic number. While a dedicated study is nec-essary to make a conclusive statement, it may bepossible to employ an argon target, which would di-rectly address the needs of the neutrino community.Measurements for helium, deuterium, and hydro-gen are also of great importance, as they would pro-vide a handle on the effect of nuclear transparencyon the exclusive cross sections and cleanly separatehadronic and nuclear effects. A scintillator targetcould also be considered. As in these cases there issome potential conflict with the dark-matter pro-gram, they may require dedicated beam time.

• In order to improve energy acceptance for low-energy charged particles, the dipole magnetic fieldcan be reduced. The effect of a reduced magneticfield on the reconstruction of higher-energy parti-cles is left to study in future work.

• Although all the generator differences discussedhere are manifest (at least in part) in the forwardregion, it would be ideal to simultaneously con-strain the hadronic energy covering also wider an-gles. This could be achieved by the combinationof LDMX data and e4nu CLAS data. It is alsopossible to install a wide-angle detector in front ofLDMX, to record both types of information at anevent-by-event level.

• Additional detector systems such as improved sil-icon tracking or high-angle scintillating detectorscould improve the angular acceptance of LDMX forelectron-nucleus measurements. Their benefits andpotential costs, including the effect on the dark-matter program, will require further study.

VIII. SUMMARY AND CONCLUSIONS

Modern neutrino experiments depend on the abilityof event-generator codes to accurately model scatteringof neutrinos of several-GeV energies on nuclear targets.This includes predicting both inclusive cross sections andthe properties of the final-state hadronic system. Thisis a very challenging problem, as both nonperturbative

11

hadronic and nuclear effects operate in this energy rangeand must be simultaneously accounted for. No ab ini-tio treatment encompassing all this physics is presentlyavailable. The task of building a reliable event generatoris thus an art as much as a science, combining a num-ber of models in ways that fairly reflect the underlyingphysics and pass a battery of experimental tests.

Given this state of affairs, direct data comparisons areabsolutely essential for validating and improving today’sgenerators. In such comparisons, electron-scattering ex-periments have a very important role to play. They com-plement what might be learned from neutrino detectorsin several important ways, among which are high eventrates and precisely known kinematics. This point hasbeen recognized in the neutrino community [37] and mod-ern event generators are built to model neutrino-nucleusand electron-nucleus interactions using common physicsframeworks.

There exists another reason why electron-scattering ex-periments are of interest to modern particle physics: theyoffer a laboratory for testing theoretical ideas about darksectors. The LDMX experiment, in particular, has beenconceived for just such a purpose and its design has beenoptimized for searching for sub-GeV dark matter withunprecedented reach. It turns out, as we argue in thispaper, that the two seemingly unrelated tasks are in real-ity highly synergistic and LDMX will provide invaluabledata on electron-nucleus scattering processes that can bevery helpful for the neutrino-oscillation program. Witha 4-GeV electron beam, LDMX would be able to probe aregion of DUNE’s scattering phase space where the eventdensity is high (cf. Fig. 10 in Appendix C), the theo-retical description is challenging, and the existing datacoverage is very limited (cf. Fig. 8 in Appendix A).

To quantify this statement, we compared predictionsof genie and gibuu, two of the leading event generatorson the market. We argued that both statistical and sys-tematic errors achievable at LDMX are expected to besignificantly smaller than the differences between the pre-dictions of these generators (cf. Fig. 9 in Appendix B).This applies not only to inclusive cross sections, butalso to measurements of specific hadronic final states.In fact, LDMX will be able to perform high-resolutionstudies of spectra and angular distributions for a vari-ety of interaction products—making use of its capabil-ity of measuring electrons, photons, neutrons, pions, andprotons—over a large geometrical acceptance with highefficiency. These measurements will improve our under-standing of hadronic physics in the theoretically chal-lenging region of transition from resonance excitationsto deep-inelastic scattering. Moreover, LDMX has alsogood acceptance and resolution of neutrons, which area crucial source of missing energy in neutrino detectors.LDMX can thus serve as an important tool in constrain-ing the neutrino-nucleus cross-section uncertainties thatplague the neutrino-oscillation program.

For all of these reasons, we strongly encourage theLDMX Collaboration to pursue detailed modeling studies

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of the scattering processes outlined in this paper and toinclude the corresponding measurements in future datataking.

ACKNOWLEDGMENTS

We are grateful to the LDMX Collaboration for usefuldiscussions and comments. We also thank the developersof the ldmx-sw code for simulations used in this study.We thank Steven Dytman for assistance with using ge-nie version 2 in electron-scattering mode, and StevenGardiner and Afroditi Papadopoulou for help with genieversion 3 and for providing the e-Ti cross section spline.We express our gratitude to Ulrich Mosel for help in run-ning gibuu. A. M. A, A. F., O. M., P. S., N. Toro, andS. W. L. are supported by the U.S. Department of Energyunder Contract No. DE-AC02-76SF00515. N. Tran issupported by Fermi Research Alliance, LLC under Con-tract No. DE-AC02-07CH11359 with the U.S. Depart-ment of Energy, Office of Science, Office of High EnergyPhysics.

Appendix A: Current data coverage

The most extensive data coverage for inclusive elec-tron scattering is currently available for the carbon nu-cleus [78]. Figure 8 shows the (ω,Q2) kinematic regioncovered by these data, compiled from Refs. [53, 82–93].The gray-scale heat map in the background representsthe expected event distribution in the DUNE near de-tector, reproduced from Fig. 1. Each colored curve rep-resents a single dataset, taken at a fixed electron-beam

12

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FIG. 9. Event distribution as a function of electron energy transfer, ω, for the scattering angles in the range 0◦–40◦ (top left),5.0◦–7.5◦ (top right), 15.0◦–17.5◦ (middle left), 25.0◦–27.5◦ (middle right), 30.0◦–32.5◦ (bottom left), and 35.0◦–37.5◦ (bottomright). Note that electrons are required to have the transverse momentum exceeding 200 MeV.

13

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14

energy and scattering angle.

Figure 8 demonstrates that—even at the inclusive leveland for carbon—there is poor data coverage where theDUNE event density is the highest. As we will see below,in Fig. 10, much of this region is dominated by resonance-excitation and DIS processes, where hadronic physics ishighly complex. For improving generator models, it isessential to have not only the inclusive cross sections, butalso exclusive measurements that record multiple final-state hadrons. Such measurements are at present notavailable.

It is important to note that, even in the context ofthe inclusive cross sections, the phase space is three-dimensional. That is, three independent kinematic vari-ables are required to fully specify the kinematics: in ad-dition to ω and Q2, a third variable—such as θe (or beamenergy)—needs to be given, in order to calculate the in-clusive cross sections. For a point in the (ω,Q2) space,a good agreement between a cross section estimate andexperimental data for some θe does not guarantee thatthe same is true for significantly different values of θe.The same applies to beam energy. Therefore, for the pur-pose of long-baseline neutrino program, currently avail-able data for inclusive cross sections cover even smallerfraction of the relevant kinematics than Fig. 8 may sug-gest at first glance. Measurements of the cross sectionsfor different scattering angles or beam energies over thesame points in the (ω,Q2) space are necessary to performan extensive validation of our description of electroweakinteractions with nucleons and atomic nuclei.

Last, we would like to acknowledge that in additionto the cross sections reported in Refs. [53, 82–93], theF2(x,Q2) structure functions for deuteron and carboncan currently be validated against the CLAS measure-ments performed over a broad kinematics [94, 95]. TheLDMX results will complement these inclusive data, aswell as provide information on (semi)exclusive cross sec-tions.

Appendix B: Inclusive electron distributions

Figure 9 shows simulations of the inclusive e-Ti crosssection for additional scattering angles θe, extending theresults of Fig. 4. We see that there is general disagree-ment between genie and gibuu predictions, at all valuesof θe. Note that due to the trigger selection, ω > 1 GeV,and the transverse momentum cut, pT > 200 MeV, onlythe events corresponding to the final electron energyE′ > pT / sin θ contribute to the distributions presentedhere.

Appendix C: DUNE event distributions

In Fig. 10, we break down the DUNE event samplesimulated with gibuu according to the individual chan-nels modeled by the generator: quasielastic (QE), meson-exchange current (MEC), resonance production (RES),and deep-inelastic scattering (DIS). The color scale isconsistent in all four panels, i.e., the same color indi-cates the same event density. The blue and green linesshow constant values of electron-scattering angles θe andtransverse momenta pT .

When Q2 ' 2Mω, M being the nucleon mass, themain mechanism of interaction is quasielastic scattering,νµ + n → µ− + p, on individual nucleons inside the nu-cleus. Accordingly, we see a linear shape in the top leftpanel in Fig. 10. At Q2 ' 2Mω + M2

res −M2, the en-ergy transferred to the struck nucleon N is sufficient toexcite a baryon resonance state Bres with mass Mres, i.e.,νµ+N → µ−+Bres. When the energy transfer increasesfurther, production of higher hadronic resonances grad-ually transitions to the DIS regime, in which interac-tions are treated at the quark level. According to gibuu,DUNE near detector events are dominated by DIS (39%)events, closely followed by QE (25%) and resonance pro-duction (24%) events.

It should be kept in mind that, physically, the bound-ary between RES and DIS events—as well as between QEand MEC events—is vague, and is a matter of conventionin a given generator.

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